In an asymptotically AdS geometry, there exist stationary black brane solutions characterized by a uniform Hawking temperature, \(T\) , and a constant center of mass velocity \(u^{\mu }\)  relative to an observer at infinity. Here greek indices \(\mu ,\,\nu ,\)  denote coordinates on the boundary of AdS space. As discussed in detail in [1]}, it is possible to perturb these black brane solutions in a derivative expansion where one assumes that derivatives of \(T\)  and \(u^{\mu }\)  are small relative to \(T\) . The perturbative solution to the Einstein equations takes the form
\(g_{ab}dx^a dx^b =& -r^2 h(r) u_{\mu }u_{\nu } dx^{\mu }dx^{\nu } + r^2 P_{\mu \nu }dx^{\mu }dx^{\nu } \\& - 2 u_{\mu }dx^{\mu } dr + \mathcal {O}(\nabla )\,,\) 
